Grade 9
  1. Number Systems  1.1. Real Numbers  1.2. Rational Numbers  1.3. Irrational Numbers  1.4. Properties of Real Numbers  1.5. Laws of Exponents and Surds  1.6. Representation on the Number Line  1.7. Decimal Representation of Real Numbers
  2. Polynomials  2.1. Definition and Types of Polynomials  2.2. Degree of a Polynomial  2.3. Zeroes of a Polynomial  2.4. Remainder Theorem  2.5. Factorization of Polynomials  2.6. Algebraic Identities  2.7. Polynomial Equations and Roots
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Plotting Points on the Plane  3.3. Distance Formula  3.4. Section Formula  3.5. Area of a Triangle  3.6. Coordinates of Midpoint and Centroid
  4. Linear Equations in Two Variables  4.1. Solutions of a Linear Equation  4.2. Graphical Method  4.3. Algebraic Method  4.4. Applications of Linear Equations  4.5. Linear Inequalities
  5. Introduction to Euclidean Geometry  5.1. Basic Definitions and Terms  5.2. Axioms and Postulates  5.3. Theorems on Angles and Lines  5.4. Properties of Parallel Lines  5.5. Construction of Triangles  5.6. Congruence Axioms
  6. Lines and Angles  6.1. Angle Pair Relationships  6.2. Parallel Lines and Transversal  6.3. Properties of Angles Formed by Parallel Lines  6.4. Angle Sum Property of a Triangle  6.5. Types of Angles
  7. Triangles  7.1. Congruence of Triangles  7.2. Criteria for Congruence  7.3. Properties of Triangles  7.4. Inequalities in a Triangle  7.5. Similarity of Triangles  7.6. Pythagorean Theorem
  8. Quadrilaterals  8.1. Properties of Quadrilaterals  8.2. Types of Quadrilaterals  8.3. Midpoint Theorem  8.4. Properties of Parallelograms  8.5. Trapezium and Kite Properties  8.6. Diagonals and their Properties
  9. Areas of Parallelograms and Triangles  9.1. Area of a Parallelogram  9.2. Area of a Triangle  9.3. Proofs of Area Theorems  9.4. Applications of Area Theorems
  10. Circles  10.1. Definitions and Terms  10.2. Properties of a Circle  10.3. Chord Properties  10.4. Tangents to a Circle  10.5. Cyclic Quadrilaterals  10.6. Arcs and Angles Subtended
  11. Constructions  11.1. Basic Constructions  11.2. Constructing Bisectors  11.3. Constructing Triangles  11.4. Constructing Tangents to a Circle  11.5. Constructing Angles of Given Measure
  12. Heron's Formula  12.1. Area of a Triangle Using Heron’s Formula  12.2. Application to Different Triangles and Quadrilaterals  12.3. Proof of Heron’s Formula
  13. Surface Areas and Volumes  13.1. Surface Area of a Cube Cuboid and Cylinder  13.2. Volume of a Cube Cuboid and Cylinder  13.3. Surface Area and Volume of a Cone  13.4. Surface Area and Volume of a Sphere and Hemisphere  13.5. Conversion of Units  13.6. Volume of a Prism
  14. Statistics  14.1. Collection of Data  14.2. Presentation of Data  14.3. Measures of Central Tendency  14.4. Mean Median and Mode  14.5. Graphical Representation of Data  14.6. Range and Measures of Dispersion
  15. Probability  15.1. Introduction to Probability  15.2. Experimental Probability  15.3. Theoretical Probability  15.4. Simple Problems on Probability

Grade 9 → Quadrilaterals ↓


Diagonals and their Properties


In this detailed exploration, we will discuss the nature and properties of diagonals in various quadrilaterals. Quadrilaterals are four-sided polygons that have interesting properties, especially when it comes to their diagonals. Understanding these properties helps us solve many geometric problems related to shapes and structures in practical applications.

What is a diagonal?

In a polygon, the diagonal is a line segment that connects any two non-contiguous vertices. For a quadrilateral, which has four vertices, the diagonals are the segments connecting opposite corners.

The basic formula for determining the number of diagonals

The formula for finding the number of diagonals (D) in a polygon with n sides is:

d = n(n - 3)/2

For a quadrilateral (n = 4):

D = 4(4 - 3)/2 = 2 diagonals

Types of quadrilaterals

Before discussing the properties of diagonals, let us briefly review the different types of quadrilaterals, as this will help us understand their diagonal properties:

  1. Rectangle: Opposite sides are equal and all angles are right angles.
  2. Square: All sides are equal and all angles are right angles.
  3. Rhombus: All sides are equal but the angles are not right angles.
  4. Parallelogram: Opposite sides are equal and parallel.
  5. Trapezoid: Only one pair of sides is parallel.
  6. Kite: Two pairs of adjacent sides are equal.

Properties of diagonals in different quadrilaterals

Rectangle

The diagonals in a rectangle have important characteristics:

  • The length of the diagonals is equal.
  • The diagonals bisect each other.
  • The diagonals form two right-angled triangles within the rectangle.

Square

Squares have the same diagonal properties as rectangles, but have unique properties because of the equilateral sides:

  • The diagonals are of equal length and bisect each other at 90 degrees.
  • The diagonals divide the square into four equal isosceles right triangles.

Rhombus

The diagonals in a rhombus provide interesting properties:

  • The diagonals are not equal but they bisect each other at right angles.
  • The diagonals act as the angle bisectors of the rhombus.
  • They divide the rhombus into four right-angled triangles.

Parallelogram

The properties of the diagonals of a parallelogram are as follows:

  • Diagonals bisect each other but are not necessarily equal.
  • The diagonals divide the parallelogram into two congruent triangles.

Trapezium

Different types of trapezoids have different diagonal properties:

  • Generally, the diagonals of a trapezium are neither equal nor necessarily bisect each other.
  • In an isosceles trapezium, the diagonals are equal.

Kite

A kite has the following diagonal properties:

  • Diagonals bisect each other at right angles.
  • One diagonal bisects the other.

Calculating diagonal length

The lengths of the diagonals in quadrilaterals can be calculated using the distance formula when the vertex coordinates are known, or by trigonometry in rectangles and squares.

Example: Length of diagonal in a rectangle

Consider a rectangle with length l and width w. The length d of the diagonal can be found using the Pythagorean theorem:

d = √(l² + w²)

Example: Square

For a square with side s, the length d of the diagonal is given by:

d = s√2

Applications of diagonals

It is important to understand the properties of diagonals because they are widely used in real-world applications. Here are some scenarios where diagonal properties are exploited:

  • Architectural design: Diagonals provide strength to structures such as bridges and foundations.
  • Structure design: Diagonals are used to create additional support and to stabilize structures.
  • Navigation and mapping: Shortest paths or routes can be provided on maps by calculating distances using diagonal properties.

Conclusion

Diagonals in quadrilaterals have a spectrum of properties that apply to things like congruence, intersection, and measurement. Recognizing these properties opens the window to a wide range of mathematical problems and solutions in both theoretical and practical fields. The simple act of drawing a diagonal on a quadrilateral reveals a multitude of geometric truths.


Grade 9 → 8.6


U
username
0%
completed in Grade 9

← Prev (8.5)
Trapezium and Kite Properties
Next (9) →
Areas of Parallelograms and Triangles

Comments 



Diagonals and their Properties

https://www.buddymath.com/g9/8.6?bmal=en